SYNTHESIS AND ANALYSIS OF EPICYCLIC GEAR TRAINS

CHAPTER I

INTRODUCTION

1.1 Brief History of Synthesis And Analysis

In Mechanical Engineering design, an engmeer uses scientific

principles, technical information and imagination in the description of a Machine

to perform a specified task. The designer uses his knowledge and imagination to

generate as many feasible solutions as possible. Out of the three important inter

related phases of design process i.e. 1) product planning phase 2) conceptual

design phase and 3) product design phase, the second phase is an important phase,

as many design alternatives are generated, analyzed and selection of most

promising choice is made for detailed designing. In this phase the skeleton form

of the machine or a part of a machine component is developed. Hence a lot of

attention is paid by the designers to this conceptual design phase of mechanisms.

The conceptual phase of design is accomplished by the designer's

intuition, ingenuity and experience. An alternative approach is to generate an

atlas of Mechanisms classified according to functional characteristics for use as

the source of ideas for Mechanism designers. This approach in general may

identify all feasible mechanisms and lead to an optimum design. In this systematic

design process, the designer uses kinematic structure representation of

Mechanisms. The kinematic structure contains the essential information about

which link is connected to which other link by what type of joint. Study of

kinematic structure of mechanisms for the understanding of their function helps in

development of enumeration, identification and classification of kinematic chains

and mechanisms. Checking for isomorphism at the conceptual stage will avoid

duplication in generation process.

Though many stages are involved in the creative phase of design,

Analysis and synthesis are two important phases. In kinematic Analysis the

relative motion associated with links of a mechanism or machine are studied, and

this is a critical step towards proper design of mechanisms. While in Kinematic

Synthesis, which is the reverse process of analysis, the designer generates

different alternative structures to satisfy the desired motion characteristics of an

output link for the given motion to an input link in relation to fixed link.

There are three phases of kinematic synthesis, viz, Type synthesis,

Number synthesis and Dimensional synthesis. Dimensional synthesis deals with

the determination of proportions of different links of a pre-decided structure in the

first two phases. In Type synthesis, designer at the conceptual design phase

considers all possible types of mechanisms and selects most suited one to meet the

functional requirements, materials, manufacturing process and cost. In the second

phase i.e. Number synthesis, designer decides about the number of links, types of

joints and number of joints needed to achieve a given number of degrees of

freedom of mechanism selected in phase one. Number synthesis also involves

enumeration of all feasible kinematic structures or linkage topologies for a given

number of degrees of freedom, number of links, and type of joints. Hence this

2

phase is also known among kinematicians as Structural Synthesis or Topological

Synthesis.

Although understanding of structural characteristics of a given type of

mechanism is critical for the development of efficient algorithms, various

methodologies have been developed for systematic enumeration of kinematic

structures. First, the functional requirements of a class of mechanisms are

identified and the structures of same type are enumerated systematically using

graph theory and combinatorial analysis. Structural synthesis of kinematic chains

and mechanism is of two stages. First one is determination of all possible

structurally distinct kinematic chains with a given number of links and specified

degree of freedom and the other one is identification of distinct inversions that can

be obtained from a given kinematic chain. Many researchers in the field of

kinematic synthesis and analysis of mechanisms had focused their attention on the

investigation of planar linkages. A substantial amount of literature is available for

planar linkages.

Freudenstein [I] in 1955, gave an appropriate but a numerical method

for dimensional synthesis of a four bar linkage. This paper is intended to make

the construction of 4-link Mechanism for the functional relationships for

computing purposes relatively easy and certain, as the design of synthesis of

linkages is generally a problem of choosing a set of arbitrary parameters, so that

the resulting configuration produces a desired motion.

3

McLaman C.W. [2] in 1963 synthesized six link plane mechanisms

using numerical analysis. He developed equations relating the link dimensions and

the input & output crank angles for Watt & Stephenson chains. An iterative

solution is explained and the example problems are solved and discussed.

Crossley F.E. [3] in 1966 made a comparison of the unpublished work

of Alt (1953) with a previous work of Crossley. The unpublished work of Alt has

a collection of ten link plane kinematic chains. While compiling the collection of

I 0 bar chains the greatest problem was to distinguish whether two arrangements,

which might appear alike, were actually the same or different and this lead to the

definition of isomorphism between linkages so that the links arrangement in a

kinematic chain with a given number of links and DOF is unique and distinct.

Thus coined is the word topological isomorphism. Two patterns form an

isomorphic pair if there is one to one correspondence, if not they are non­

isomorphic.

In 1966 Davies T.H. and Crossley F.E [4] obtained the censuses of

seven, nine, ten and eleven bar kinematic chains. They state that at early stage in

design process the question of dimensions is largely irrelevant both in respect to

distances between pairs and cross sections of the bodies they connect. Though

Reuleaux [5] devised a comprehensive symbolic notation capable of describing

kinematic relationships, his notation is rarely used because the graphic qualities of

drawing are entirely lost. A graphic representation of a kinematic chain is more

4

desirable as the number of links increases. Denavit & Hartenberg [6] worked out

a matrix fonn-taking cognizance of the different fonns of joints in the chain.

In 1967 L.S. Woo [7], gave Type synthesis of plane linkages. He

enumerated plane kinematic chains (PKC) having ten links. He also gave an

algorithm for deriving all PKCs from those of lower number of links. He for the

first time gave all the 230 graphs of ten links and one DOF kinematic chains

(KC).

Freudenstein F [8] in 1967 presented the basic concepts of Polya's

theory [9]. Polya's Hauptsatz is stated without proof using permutations, groups

and graphs and its use is illustrated with reference to the structural classification

of mechanisms. He presented the elementary aspects of theory needed for the

enumeration of graph structures, which can be defmed as graphs.

Davies. T in 1968 [10] extended the Manolescu's classification of

PKCs and mechanisms with mobility greater than or equal to one using graphs

theory. These extensions enabled general theorems to be presented that concern

the structure of many kinematic chains including those having mobility one.

Turner J. in 1968 [ 11] gave generalized matrix functioning and

studied the graph Isomorphism. He proves that a graph is not characterized by the

Eigenvalues [12, 13, 14] of its Adjacency Matrix and gives several non-isomorphic

graphs with the same Eigen values. He therefore gives a set of Matrix functions

also known as "lmmanants" to characterize graphs for isomorphism.

5

Haas S.L. and Crossley F.R.E. [15] in 1969 used number synthesis of

linkages to produce a collection of linkage forms with mobility four. The

procedure explained in four steps and to describe the same Frank's notation is

used.

In 1971 Manolescu N.J. [16] described a method of linkages census

based on transforming of Baranov Trusses into PKCs using graphisation.

Kinematic chains with m multiple joints and simple links (KCMJSL) obtained by

using dyad amplification (DA), with greater number of links but with same DOF.

This paper was presented in memory of Baranov, Dobrovolski and Artobolevski,

the founders of the modem theory of mechanisms. Baranov Truss (of zero DOF)

is a structural framework composed of pin-jointed bars or beams related by 3 *1 -

2 * j = 3 where l is number of bars and j is number of simple joints.

Huang M and Sony A.H [17] in 1973 used graph theory and Polya's

theory of counting to synthesise and analyze structures of planar and three­

dimensional kinematic chains. They gave a mathematical model to perform,

structural analysis and synthesis of PKCs with kinematic elements like revolute

pairs, cam pairs, springs, belt- pulleys, piston - cylinder and gears, they

enumerated eight link KCs with above kinematic elements. They also developed

a model for analysis and synthesis of multi-loop spatial KCs with lower and

higher kinematic pairs.

Artobolevskii I.I [18] in 1974, while addressing the 131h ASME

Mechanisms Conference, New York besides other aspects stressed the need for

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creating new mechanisms, automatic machines for the automation of manual and

intellectual labour of man. He underlined the necessity of designing complex

reduction gears with planetary and differential schemes.

In 1974, J.J. Uicker Jr and A. Raicu [19] presented a method for

determining a set of identification numbers for a kinematic chain and thereby

using these numbers to detect isomorphism in kinematic chains. These authors

present a procedure based on graph theory, which leads to a numerical algorithm

for testing for isomorphism of two kinematic chains. Though counter examples

are found to this method by later researchers [26,29] this paper is a pioneering

work in the field of analytical isomorphism of kinematic chains.

A.C.Rao [20] in 1975 introduced moment concept to kinematic chains

and used this moment method for the analysis of planar complex mechanism. This

is a very simple method for the velocity and acceleration analysis of complex

mechanisms.

Gred Kiper and Dieter Schian in 1975 [21] gave a computer-assisted

synthesis of 12 link Grubler kinematic chains. They used linear graphs for

synthesis of these 12 link chains.

A.C.Rao in 1977 [22] illustrated with an example that a five bar single

loop linkage can be used as mechanism with single input, even though a five bar

single loop chain is a two DOF chain.

In 1978 [23] gave an accurate analysis of slider crank mechanism. He

underlined how the performance of a linkage is influenced by the elasticity of

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hnks and JOmt dearances. He also stresses that the assumption of rigid link is not

vahd when the mechanism designed based on the above assumption, is operated

under high static and inertia forces.

In 1979 [24) described an easy and alternative method for the design

of spatial slides crank mechanism for function generation using minimum

standard deviation as criterion for closure error. The method is illustrated by a

numerical example with six precision points.

Mrutyunjaya T.S. and Raghavan M.R. in 1979 [25) presented a

method based on Bocher's formulae for determination of characteristic

coefficients, which are indices of isomorphism in kinematic chains. They gave a

physical meaning of these coefficients and presented algebraic test for

determining whether a chain possesses total, partial or fractionated freedom [ 1 0].

They also gave generalized Matrix notation to represent and analyze multiple

jointed chains.

T.S. Mruthyunjaya [26] in 1979 synthesised kinematic structures by

transformation of binary chains. This can be used to derive all possible simple

and multiple jointed chains of positive, zero or negative DOF. The method is

illustrated by applying the theory to the case of chains with DOF -1,0,1 & 2.

M. Kothari and A.C.Rao [27] in 1980 synthesised two DOF slider

crank Mechanisms for minimum structural error. The synthesis equations of

seven links two DOF Mechanisms are highly non - linear. Hence special

techniques are required for this solution. They used an epicyclic gear train driven

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slider crank mechanism with flexibility-attached slider to generate functions of

two variables. The resulting linear displacement equations are convenient for

closed form synthesis of the mechanism.

Mayourian M and Freudenstein F in 1984 [28] developed an Atlas of

kinematic structures of Mechanisms using graphs theory. The Atlas contains 35

graphs, which define the kinematic structures of a wide class of plane and three­

dimensional mechanism with up to six links. They have shown that 4000

mechanisms can be obtained from these 35 graphs with turning, prismatic and

gear pairs and having one DOF.

T. S. Mrutyunjaya in 1984 [29] gave a computerized methodology for

structural synthesis of kinematic chains in two parts. In part one he presented the

formulation of a methodology which led to the development of a computer

programme for the structural synthesis and analysis of kinematic chains with

simple joints and DOF more than zero. While, in part two he established the

reliability the above computer programme for structural synthesis and analysis of

simple jointed kinematic chains by applying the programme to several cases such

as ?link zero freedom chains, 8 and 10 link single DOF chains.

T.S. Mruthyunjaya and M.R. Raghavan in 1984 [30) used link-link

incidence Matrix to represent simple jointed kinematic chains. Algebraic methods

are developed to determine structural characteristics like - type of freedom of a

chain, the number of distinct linkages and inversions that can be derived from a

9

chain. Graph theory is used for the computer-aided analysis of kinematic

structures. Several typical examples are presented to supplement the theory.

In 1984 A.K. Khare and A.C.Rao [31] used reliability concept for

structural error synthesis of mechanism. This method is simple and leads to

closed form solution. The mechanism designed by this method can perform with

any desired reliability. Reliability of a system can be defined as the probability

that the system will perform intended design function satisfactorily under

specified conditions. A numerical example is given to illustrate the method and

results are compared with those available. This concept can be extended to the

design of other mechanisms like cam mechanisms without any difficulty.

In 1984 [3 2] synthesised a slotted link with a flexibility-attached

slider with seven precision points for path generation. The displacement

equations obtained are linear and give closed form as well as optimum design of a

mechanism. The given equation is in a convenient form for the application of the

least square methods.

In 1985 [33] gave a method to select the input and output links in

distinct mechanisms under the study of structures analysis and syntheses of

kinematic chains and mechanisms. The method is based on the minimum number

of joints among the input and output links. This concept is applied to two DOF

links. Method is illustrated with seven and nine link chains as examples. The

concept can be extended to linkages with larger number of links having higher

DOF, fractionated DOF and input and output on floating links.

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In 1985 [34) presented a method to overcome the shortcomings in

general graph theory and reflect the node and circuit properties. Logarithmic

functions or entropies, which can be used to detect isomorphism and get a clear

idea of the quality of motion transmission, are derived using circuit and cut-set

matrices. Entropy of a coefficient Matrix for a kinematic chain is an invariant and

becomes a property of the chain.

Wayne J. Sohn and Freudenstein Fin 1986 [35] applied dual graphs

to the automatic generation of the kinematic structures of mechanisms. A

powerful new representation of the kinematic structure of mechanisms has been

developed that permits a highly efficient, completely automatic procedure for the

computer-generated enumeration of the kinematic structures of mechanisms. The

kinematic structures of one, two and three DOF planar linkages with up to four

independent loops have been enumerated. The path generated by a mechanism

deviates from the specified path and the deviation is expressed by the reliability

index.

In 1986 R.P. Sukhija and A.C.Rao [36], used reliability index for

optimal synthesis. Tolerance is also allocated on link lengths using the concept of

reliability index. The reliability is maximized for optimal synthesis of

mechanisms. Also this synthesis method gives a closed form solution, which is a

great advantage over iterative methods of formulating an objective function and

minimizing it.

I I

Many space mechanisms can be fonned with four links and joints

having different DOF. No measure was available to know which of the available

mechanism possess greater mobility or flexibility. Flexibility is different from

DOF. Though the mobility of a chain increases with the number of links, one is

not sure how the topology of links, types of links and joints, their number and

sequence influence the mobility. A.C.Rao in 1986 [37] combined graph theory

with the concepts of probability and developed simple equations to investigate the

relative merits of planar and spatial kinematic chains. Greater the flexibility or

mobility, higher is the ability of a kinematic chain to meet the motion

requirements. Entropy corresponds to the total connectivity of each link and for

given entropy; linkages in which all the links have equal corrnectivity have greater

flexibility.

Distinct ten-link kinematic chains are identified by C. Nageswar Rao

& A.C.Rao in 1986 [38], using an invariant obtained by the criteria of shortest

path through which motion is transmitted from joint to joint. The number of flow­

links between various joints may be arranged for a mechanism in the form of a

Matrix. The total of each row in the matrix is obtained and then the sum of all

such totals. MNL scheme is used to obtain the invariant and can be used to detect

isomorphism.

An assessment of merits and demerits of available methods for

detection of isomorphism in kinematic chain is presented by T.S. Mrutyunjaya

and H.R. Balasubramanian in 1987 [39]. Another test based on the characteristic

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coefficient of the degree matrix of a graph is proposed to detect isomorphism in

kinematic chains. The test is successfully applied to simple jointed kinematic

chains with single DOF and with up to ten links, two DOF chains with up to nine

links and three DOF chains with up to ten links.

Ambekar A.G. and Agrawal V.P. in 1987 [40], used min code for

canonical numbering of kinematic chains and isomorphism problem. It explains

the concept of min code and discusses its properties relevant to kinematic chains.

The algorithm given is based on the method available in graph theory literature in

chemistry. Min code is unique and is suitable for testing isomorphism in

kinematic chains. The code can be decoded positively hence possible to use in

storing and retrieving of the kinematic chains and mechanisms.

Ambekar A.G. and Agrawal V.P. in 1987 [41), used min code as

canonical number to give a unique number for kinematic chains with simple

joints. A method is suggested to identify mechanisms, path generators and

function generators through a set of identification numbers. Min code is also

shown to be effective in revealing the topology of kinematic chains and

mechanisms with different types of lower pairs and or simple and multiple joints.

Pseudo-Hamming distance of a kinematic chain is an invariant and

hence it can be associated with some structural property of the chain. Hamming

distances are sensitive to changes in the type & number of links and loop

formation. This aspect is studied in 1988 [ 42] and proves that higher the

Hamming value the better is structural error performance of the chain. For chains

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with the same connectivity, the chain having greater Hamming value contains

joints of greater DOF. Chains having different Hamming value have different

capabilities even though the two chains have same connectivity.

The number and type of loops and their arrangements in the chain

may be such that the chain may have total or partial or fractionated freedom. The

methods reported to that date were not very efficient computationally. In 1988

[43] presented a simple and computationally efficient method using Hamming

distances to detect isomorphism, inversions and type of freedom in multi - DOF

and multi loop mechanisms.

Isomorphism among kinematic chains with sliding pairs was studied

by A.C.Rao & C. Nageswar Rao in 1989 [44]. This method is based on the

concept of equivalent chains and the method is an improvement on the method

given by Uicker and Raicu.

In 1990 [45] made a companson of linkage mechanisms usmg

Hamming number method. Selection of ground, input and output links for the

specified task like path or function generation can be made. It is explained with

examples of single and multi - DOF mechanisms. Comparison of inversions is

also made.

Hwang W.M. and Hwang Y.W in 1992 [46] presented a computer­

aided structural synthesis of planar kinematic chains with simple joints. This

paper consists of systematic generation of contracted link adjacency matrices,

detection of degenerate chains and identification of isomorphic chains. The

14

programme used by the authors automatically synthesises planar kinematic chains

with the given number of links and DOF and they have listed kinematic chains

with up to 13 links. They obtained 6862 distinct and non-degenerate kinematic

chains with 12 links and one DOF as against the earlier result of 6856 kinematic

chains given by Kiper and Schian [ 4 7]. They state that characteristic polynomial

of degree matrix given by Mrutyunjaya and Balasubramanian [39] is successful in

distinguishing the structure of an the kinematic chains with up to I 0 links.

However it fails to distinguish the kinematic chains with 12links and one DOF.

Two linkages with the same number and type of links but with

different topology, when optimized for the same task win behave differently from

the viewpoint of dynamics. This is due to difference structure. In 1992 [48]

studied the structure dependent dynamic behaviours of linkages. Comparison of

linkages based on structure for dynamic performance can be made at the pre

design stage. It is shown that a degree of non-linearity exists between the input

and output links of a kinematic chain depending on the structure. Accelerations of

the links in the chain are correlated to this non-linearity. Validity of the method is

proved by taking six bar and eight bar chains as examples. A chain which

possesses good dynamic behaviour need not be good from viewpoint of static

behaviour.

A.C.Rao and C. Nageswar Rao in 1993 [49] made a comparison of

performance of Robotic structures. The accuracy, ease of control, size of working

space, computational effort and accessibility depend upon the structure of the

IS

open chain. A numerical approach using linkage adjacency matrices is used to rate

the linkages for their relative abilities.

Varada Raju et al in 1994 (50) gave a method for structural synthesis

of simple jointed PKCs. They presented a direct, quick and reliable method to

synthesise planar simple jointed chains, open or closed with single or multi - DOF

and with any number of links. A simple, concise and unambiguous notation is

used to represent a chain.

In 1997 [51] used Hamming number Technique to generate planar

kinematic chains. The Hamming number Technique is extended to reveal identity

and symmetry among the links and joints of chains. This in turn enables

generation of distinct chains without having to test for degeneration and

isomorphism. Formulae are given to calculate the number of such chains using

this method. Only a few chains generated needed to be tested for isomorphism.

A.C.Rao and Jagadeesh Anne in 1998 [52] presented quantitative

methods, based on the topology of chains, in order to compare all the distinct

chains with the specified number of links and DOF. In this work topology based

characteristics of kinematic chains like work space, rigidity, input joint and

isomorphism in kinematic chains are dealt with. Compact linkages do not have

greater workspace. Using distance and self-loop concepts of graph theory one can

decide the joint at which motion is to be supplied to get best result from the input

link.

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In 1999 [53] discussed a method based on the moment concept to test

isomorphism among kinematic chains and inversions. The numerical strings

represent kinematic chains uniquely. Symmetry in kinematic chains is studied

and numerical measures are proposed to quantify symmetry in PKCs. Multi -

DOF kinematic chains are compared for parallelism so that selection of a best

chain for a robot can be made. The proposed theory is applied to six bar and eight

bar planar kinematic chains to illustrate the method.

Multi - DOF PKCs can be considered for application as in-parallel

robots in view of their greater rigidity. Since a number distinct chains are

available with the same number of links and DOF for consideration as in-parallel

robotic structures. In 2001 [54], presented a simple and logical method to decide

which of these chains is more in-parallel so that chain selected fulfills the

specified task such as workspace, rigidity. Measure of parallelism developed is

also used to compare distinct chains for efficiency and component velocities.

Robot arms are open chains and each joints is actuated independently.

These suffer from disadvantages like less rigidity, accumulation of mechanical

errors from shoulder to end effects, control problem etc. Platform type robots are

an alternative to open chain robot arms. In 2001 [55] proposed numerical

measures to compare all the distinct planar linkage mechanisms at the conceptual

stage of design. The method presented is based on the deployment of design

parameters in different topological features of the chains viz., links, joints and

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loops. The theory is applied to two - DOF nine link chains and three - DOF ten

link chains.

A.C.Rao and P.B. Deshmukh in 2001 [56] gave a computer aided

structural synthesis of planar kinematic chains obviating the test for Isomorphism.

Link assortment of a kinematic chain is an orderly sequence of the number of

binary, ternary, quaternary etc. links that are needed to form a chain. Important

feature of link assortment is total number of links with connectivity 3, 5, 7 etc

cannot exist in odd number with other links of even connectivity. The steps

involved in arriving at the kinematic chains are simple and straightforward and

allows the designer to visualize the generation throughout the process. The type

of freedom, full, partial etc can be known right at the time of generation. Also a

lot of saving in computer space is accomplished.

1.2 Review of Literature on Planetary Gear Trains

It is a known fact that these techniques used for the structural

synthesis of planar linkages can be extended to other types of mechanisms such as

Gear drives, Cam mechanisms, Hydraulic piston-cylinder mechanisms, spring

mechanisms etc,. However studies dealing with such extensions or development

of techniques suitable for structural synthesis of these specific types of

mechanisms are quite limited. Recently, the attention of the researchers ha shifted

to the structural synthesis of geared kinematic chains. Nearly all branches of

industry require reliable mechanisms for variable transmission. Of different

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available means of Power transmission from one shaft to another, gear trains

constitute one of the best methods, because gear trains (gears) represent a high

level of engineering achievement besides rolling contact bearings. There are

enormous variety of possible gear train configurations for such application as

automobiles and helicopters and differentials, gas turbine-engines, machine tools,

robotic wrist mechanisms, steering mechanisms for track laying vehicles etc,.

A gear train is referred as an "ordinary gear train" if the shaft axes are

fixed in space throughout their motion. Since for the given number of gear

elements and size, the gear ratio is less for an ordinary gear train, Epicyclic or

Planetary Gear Trains (EGT or PGT) are used to achieve the same or higher gear

ratio with less number of elements. An Epicyclic Gear Train (EGT) consists of

one or more central sun gears with gears in mesh with them revolving around

them like planets of sun, giving Epicyclic motion to the planets. Each planet is

associated with a link called Gear Carrier or Arm, which ensures the constant

center distance between two gears in mesh.

A Kinematic Chain with gears as elements, from which an Epicyclic

drive can be obtained by keeping one link fixed is also known as a Geared

Kinematic Chain (GKC) or an Epicyclic Chain (EC).

Because of the inherent advantages of lightweight, compactness,

differential drive and above all high-speed ratios possible with limited number of

elements, Planetary gear trains are often used in transmission systems.

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In the earlier stages of study of PGTs the attention is towards their

analysis. Levai (1968), Colbourne ( 1972), Townsend (1972), Mihal (1978),

Wojcik (1978), Willis (1982) and Wilson et al (1983) used Vector Analysis [57]

to study the PGTs. Martin (1982), Wilson (1983), Tye and Cleghorn (1985 and

1987) used Table Method [57] for the analysis of PGTs. Train Value method [57]

is used by Tailai (1983), Pazak et al (1984) Bacgi (1987) and Freudenstein

(1971)) Freudenstein and Yang (1972), Tsai (1985), Hsieh et al (1988), Cheng­

Ho Hsu and Kin-Tak Cam (1992) used graph theory method [57] for the

kinematic analysis of PGTs.

In 1968 Levai [58] has described in his publication how all-34 types

of PGTs can be derived from one general fonn of the single gear train. He has

also given a family tree of planetary gear trains. Though analysis of planetary gear

trains has spanned over three decades from 1968, the synthesis of planetary gear

trains in a systematic manner started in 1979 by Buchsbaum and Freudenstein

[59]. They used network concepts and combinatorial analysis to develop methods

for the enumeration of PGTs according to kinematic structure. They illustrated

that separation of kinematic structure from functional considerations is useful at

the conceptual stage of design and in identifying the potentially useful PGTs.

They also introduced three different ways of representing PGTs, i.e., Schematic,

Functional and Graph representations.

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1.2.a. FUNCTIONAL REPRESENTATION

The functional representation of an epicyclic gear train refers to the

conventional schematic structure drawing of the mechanism. When analyzing the

kinematic structure or motion of planetary gear trains, the schematic diagram is

commonly used to represent the relationship among the links and joints of gear

train. The simplest epicyclic gear train is shown in fig. ( 1.1 ). The functional

representation of a planetary gear train is shown in fig (l.la). The elements

labeled as P and S are gears meshed together and A is an arm carrier.

1.2.b. GRAPH REPRESENTATION

For the systematic synthesis of epicyclic gear trains, a graph can be

used to represent the structure of a gear train. In a graph a vertex represents a

mechanical element and edge represents a joint. The edge connection between

vertices corresponds to the pair connection between links. An epicyclic gear train

has two kinds of joints one is a turning pair which connects the axis of a gear and

a carrier, and the other is a gear pair between two meshing gears. In order to

distinguish a turning pair connection from a gear pair connection, turning pairs are

represented by line and gear pairs by double line. Furthermore, the thin edges are

labeled according to their axis locations. A graph made of this and heavy edges is

some times referred to as a bicoloured graph since the two different edges can also

be represented by two different colour codes. A bicoloured graph without labeling

its thin edges is called an unlabelled graph. Fig (1.2) shows the graph

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representation of the mechanism shows in fig where vertices 1,2 and 3 correspond

to links 1,2 and 3 the thin edges 1-2 and 1-3 correspond to the turning pair

connecting links 1 and 2 and 1 and 3 the edge labels a and be correspond to the

joint axis locations a-a and b-b and the heavy edge 2-3 corresponds to the gear

pair connection between links 2 and 3 respective.

1

2 ... ___ .3

Fig (1.2)

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1.2.c. ROTATIONAL GRAPH REPRESENTATION

A rotational graph consists of gear pair edges and incident vertices

and each edge is labeled with a symbol of the transfer vertex associated with the

edge which represents the carrier of two meshing gears fig shows the rotational

graph of the mechanism shown in fig where the edge 2-3 is labeled with the

number of transfer vertex. The transfer vertex is the vertex, which is not incident

with geared edge.

1

21 ~

Fig. (1.3)

According to Freudenstein the rotational graph of a planetary gear

train is defined as the graph obtained by deleting the turning edges of the transfer

vertices from the structural graph and labeling each geared edge with the symbol

for the associated transfer vertex. This is shown in figure (1.3). According to Ravi

Shankar.R and Mruthyunjaya the defmition of rotational graph is obtained from

the structural graph by deleting the turning edges and joining the end vertices of

each geared edge directly too the associated transfer vertex with new turning

edges.

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Displacement graph is obtained by labeling the new turning pair edges

of rotational graph according to their axis locations in space. All the turning edges

forming a circuit must share a common levels in some cases a rotational graph

may contain more turning edges than the structural graphs from which it is

derived and hence may not represent a feasible structural graph. Moreover several

structural graphs may correspond to identical displacement graph. These facts

result in a difficulty in detecting the displacement isomorphism of planetary gear

trains.

The term isomorphism is defmed as the one to one correspondence

between the graphs i.e., the two graphs though they have different structures,

represent the same motion. They also gave five levels for the structural

classification of PGTs. In Graph representation of a GKC the vertices in the

Graph are links and the Edges are Pairs between the elements. Also because of

different types of pairs between various elements in the graph Hi-coloured Graph

representation is used.

In 1971 Freudenstein [60] used Boolean algebra to investigate the

kinematic structure of PGTs. He has given the correspondence between the graph

representation of PGTs and form of the displacement equations. He gives a

Canonical graph representation and determination of algebraic displacement

equations by inspection from kinematic structures. Also the same can be applied

to dynamical equations and sketching of the PGTs. Graphical representation of

24

PGTs is clearly explained. The fundamental rules of graphs of PGTs are given by

Freudenstein and Buchsbaum [59] and are elaborated by Freudenstein [60]. He

also introduced the concept of Transfer Vertex and gave its significance. He has

also given a method for the determination of transfer vertices in a graph of PGT.

He describes linear displacement equations and rotational displacement equations.

He introduced the concepts of Rotation Graphs and Rotational Isomorphism.

Rotation graph of a PGT is obtained by deleting the Turning Pair Edges and the

labeling the each geared edge with its corresponding transfer vertex. Transfer

vertex is the vertex in a fundamental circuit incident with only turning pair edges

and all edges on one side of the transfer vertex are at the same level and edges on

opposite side are at different level. He also defined the Pseudo Isomorphism in

PGTs. Seventeen non-isomorphic single Degree Of Freedom (DOF) epicyclic

chains with up to three gear pairs i.e. five elements are listed. Their corresponding

functional (Levai Notations) representations are also given.

In 1972, D.J.Sanger [61] described the techniques for the structural

and numerical synthesis of multi speed planetary transmissions. The structural

characteristic of PGTs are tabulated and a numerical example is given to illustrate

the use of the methods.

Ravisankar and Mrutyunjaya [62] in 1985 have given a fully

computerized method for synthesis of structures of PGTs. In this work the GKCs

are represented by graphs using the procedure given by Buchsbaum and

25

Freudenstein (59]. The graphs in tum are algebraically represented by Vertex­

Vertex incidence matrices on the lines similar to that given by Uicker and Raicu

[ 19] for planar kinematic chains. Various useful concepts introduced and results

developed by earlier researchers in GKCs are combined with certain concepts in

the area of planar kinematic chains for systematic computerized synthesis of

structures of geared kinematic chains. A stepwise procedure for computerization

of the synthesis procedure for PGTs is given and the computer programme is

applied to single degree of freedom geared kinematic chains with up to four gear

pairs i.e. six elements. A different definition for rotation graph is given. The

results are in concurrence with earlier published work (59,60]. All the possible

non-isomorphic rotational graphs and non-isomorphic displacement graphs are

listed. A procedure is explained to label the turning pairs in each graph to obtain

different non-isomorphic single degree of freedom {displacement) geared

kinematic chains with up to four gear pairs. Twenty-seven distinct graphs and

eighty functional diagrams are given with levels.

L.W.Tsai (63] in 1987 extended Linkage Characteristic Polynomial

method given by Uicker and Raicu [19] for the topological synthesis of EGTs of

one degree of freedom with up to four gear pairs. PGTs are represented by hi­

coloured graphs, thin edges for turning pairs and heavy edges for gear pairs.

Graphs with N elements are generated from (N-1) elements by recursive method.

In recursive method of generation, each time number of vertices is increased by

26

one. This new vertex is joined to different other vertices by one turning pair and

one gear pair in different possible ways. The definition of Linkage Adjacency

Matrix for a kinematic chain given by Uicker and Raicu [19] is modified to

include gear pairs in the graph. Accordingly the GKC adjacency matrices are

written with the following rules.

Each element in an adjacency matrix of a GKC a [i, jl is given by

a [ i, j I = 1 if a vertex i is connected to another vertex j by a turning pair,

a I i, j I= g if the pair is a gear pair

and a [ i, j I = 0, otherwise.

Further a[ i i I =0 .

The linkage characteristic polynomial p (x, g) of a PGT is given by

the determinant of the matrix (X I-A) where X is a dummy variable, I is unit

matrix and A is the adjacency matrix. Random number technique is used to

compute the value of a linkage characteristic polynomial and check for

isomorphism in graphs of GKCs of one DOF and with up to six elements. 26 non­

isomorphic rotation graphs and 80 non-isomorphic displacement graphs are given

for PGT with six links and one degree of freedom. Different ways of leveling the

turning edges also given.

27

Kim and Kwak (64] in 1990 applied Edge Permutations Technique for

structural synthesis of PGTs. Edge permutations which are induced from the

symmetric group of vertex permutations are used as mapping functions to check

isomorphism in graphs of PGTs. The result of the enumeration of graphs of PGTs

with up to seven elements and one degree of freedom are presented. The entire

procedure given is computerized without interactive job. While selecting a graph

from isomorphic graphs for next level, a graph with maximum number of ways of

labeling is selected. Label at a turning pair edge indicates the spatial location of

the axes joining its elements. Therefore isomorphic graphs with different edge

labeling have different mechanical structures. In their work there is concurrence

of results for graphs of PGTs with up to 6 links. For graphs of PGTs with 7 links

the generated graphs are 780, rotation graphs are 1089 and rotationally non­

isomorphic graphs are 144. The resulting non-isomorphic labeled graphs listed are

642.

Olson et a! [65] in 1999 dealt with Topological Analysis of single

degree of freedom of planetary gear trains. He gave a new graph representation,

which is useful in specification of input and output links. The coincident joint

graph representation makes it possible to easily determine the number of distinct

kinematic inversions of a P.G.K.C and to recognize whether or not there exists

feasible input output links.

28

lnl993 J K Shin and S. Krishna Murthy [66] developed an efficient

solution procedure for the enumeration of Epicycle Gear Trains based on the

standard code technique for bi-colored graphs. The standard code for a graph

consists of three separate codes, namely the codes of parent graphs, the vertex

codes and edge codes. The parent graph is obtained by replacing all the heavy

(geared) edges with thin edges. Isomorphism in displacement graphs is detected

efficiently using the standard code technique. They have also given the step-by­

step procedure. From canonical numbered graphs, the edge sets with in which

levels are the same are identified. These sets are rearranged in the ascending order

starting from the smallest edge number with in each set. The edges are leveled

with the first set as 1 and second set as 2 etc. The levels of each edge are

concatenated in the descending order of the edge numbers to obtain the level

codes of the displacements graphs.

A colored graph with a symmetry of "m" will result is almost "m"

distinct level codes. The three stages of non-recursive generation scheme are

explained, they are enumeration of parent graph, distributions of gear pair edges

to generate bi-colored graphs from parent graphs and verifying are bi-colored with

fundamental rules of GKCs.162 non-isomorphic rotation graphs are reported in

this study for one DOF GKCs with seven elements or five gear pairs which is

totally different from that given in [64].

29

Cheng-HO HSU and Kin-Tak Lam [67] in 1993 studied the kinematic

structure of Planetary Gear Trains with any number of degree of freedom. Graphs

are used for canonical representation of structures of PGTs by Displacement and

Rotation graphs. A single identification number used to test displacement

isomorphism in PGTs. They gave a method to identify non-fractionated multi -

DOF PGTs from their rotational graphs. In this paper an algorithm is given for the

automatic analysis of kinematic structure of PGTs. Olson Erdman and Riley [65]

in 1987 and 1988 proposed coincident joint graph for canonical graph

representation of PGTs. Hsu and Lam (69] in 1989 gave a new graph

representation for PGTs. ln 1989 Tsai and Lin [70] presented a method for the

identification and enumeration of kinematic structure of non-fractionated two

DOF PGTs. Hsu and Lam in their paper used the above concept for automatic

analysis of kinematic structure of Planetary Gear Train with any number of degree

of freedom. A Simpson gearbox shown in figure (1.4) is represented by a graph as

shown in figure (l.4a) and the corresponding new graph is shown in figure ( 1.4b).

In new graph representation all the joints at the same level are

represented by a multiple joint polygon instead of simple joints as in graph

representation used by other authors. This avoids the formation of Pseudo­

isomorphic graphs at the generation stage. For automation, the graphs are

represented by vertex-vertex adjacency matrices, both for displacement and

graphs, using definite rules given by them. He also gave linear displacement

30

equation and rotational displacement equations. Characteristic polynomial concept

used by these authors too for identification of isomorphism in graphs on the same

lines as L.W. Tsai [63]. These too used random numbers to evaluate the value of

determinant of adjacency Matrices of displacement and rotation graphs. The flow

chart of the Automation algorithm is given in seven steps. Utility of the computer

programme is demonstrated by taking some examples

A.C.Rao and J Anne [72] in 1996 studied topological characteristics

of planetary gear trains. A simple method to detect isomorphism among PGTs is

explained and best possible rates of PGTs based on their topology is explored.

Guidelines required for selecting a best possible gear train from an atlas of PGTs

are given. Comparison of characteristics like velocity ratio, loss of motion and

power is made. Rating of the inversions is also made based on edge values or

connectivities of vertices.

Goutam Chatterjee and L.W.Tsai [73] in 1996 gave an enumeration of

epicyclic gear trains used in automatic transmission. Most automatic

transmissions employ one type of epicyclic gear trains to achieve proper

equilibrium, between power and torque produced by an engine and demanded by

the road wheels. A configuration of a PGT is selected to have desired speed ratios

and meet other kinematic and dynamic requirements. In this work a systematic

methodology is formulated using canonical graphs to systematically enumerate all

possible configurations and to identify Kinematic structural characteristics of

epicyclic gear trains. The concept of automorphism is introduced in this paper.

31

Jo t Ic or

6 LJ l

-

4

a 6 -

~ 5

T

T

Fig (1.4) Fig (1.4a)

4

5

Fig (1.4b)

32

By permuting the nwnbering of vertices in an unlabeled graph,

different isomorphism graphs produced. But certain permutations produce graphs

whose corresponding vertices bear the same nwnber as the original one. These

graphs are called automorphic graphs.

Cheng Ho Hsu and Jin-Juh Hsu [74] in 1997 gave a methodology for

structural synthesis of PGTs. Acyclic graphs are used for the Synthesis of

kinematic structures of a Gear Kinematic Chains (GKC). A systematic procedure

is given for enwneration of N-vertex Acyclic graphs and generation of N-vertex

geared kinematic chains by adding (N-f-1) geared edges to each N-vertex Acyclic

graph. Structural codes are used to detect isomorphism in GKCs generated. They

assert that there is no concurrence in the result given by different researchers

regarding the total number of non-isomorphic GKCs with more than six elements.

The results given by these authors for one-DOF GKCs with up to six elements are

in complete agreement with those of earlier researches. For 3 elements the number

of GKCs is one, for four elements the number is three, for five elements the result

is 13 GKCs and for six links the resulting GKCs are 81. However Kim & Kwak

[64] enumerated 642 displacements graphs with seven elements. Hsu & Lam [67]

enumerated 636 one DOF seven element graphs. Shin and Krishna Murthy [66]

listed 659 GKCs with seven links and one DOF. Hsu & Hsu [74] listed 647 one

DOF graphs with seven elements. The catalogue of 647 one DOF GKCs with

seven links is incorporated.

33

A C Rao and A. Jagadeesh (75] in 1998 out lined Hamming number

technique and proved the reliability of Hamming number Technique by using it to

detect isomorphism in Planetary Gear Trains. Other benefits of Hamming number

technique like identifying distinct inversions are also explained.

A C Rao (76) in 1985 applied Information theory to synthesise Planar

Kinematic Chains. Structural errors that exist in linkages are taken as continuous

random variables and an entropy function is formulated in terms of design

parameters. Minimization of maximum entropy leads to optimum design

parameters that assume accurate performance.

A.C.Rao and D. Varada Raju [77] m 1991 developed Hamming

number Technique to defect isomorphism among kinematic chains and inversions.

The connectivity Matrix of a linkage, with zeros and ones, is formed and from that

using the rules given by them [23) Hamming number matrix is written. The

linkage Hamming sting is obtained by concatenating the linkage Hamming

number and arranging in descending order Hamming numbers of different links in

the linkage. A number of authors (48-56, 75-79] borrowed this Technique to test

isomorphism of planar linkages of different DOF and with any number of links.

The hamming matrix also reveals at a glance how many inversions are possible

out of a given chain.

B.P.Rao (78] in his doctoral thesis discussed the concept of structural

symmetry in civil structures. He defmed the term symmetry in graphs. Hamming

number matrix developed by AC. Rao and Varada Raju [77] is used to study the

34

symmetry in structures with zero and negative DOF. He has also shown the utility

of Hamming Matrix of a Graph to know the Structural symmetry in Chains.

In 2003 [79], has given a genetic algorithm for testing isomorphism in

generation of epicyclic gear trains. Quantitative measures are developed in a very

simple way using principles of genetic algorithms. These measures are utilized to

list isomorphism in PGTs and to know the characteristics like speed ratios and

transmission efficiency in a comparative sense. This will help to know the merits

and demerits of GKCs with out having to actually design fabricate and test them

for the required performance.

35

1.3 MOTIVATION BEHIND SELECTION OF THIS TOPIC

Thorough study of available kinematic literature reveals that

computerization of structural synthesis of kinematic chains has received much

wider attention. Also a large number of studies have been reported in literature

concerning isomorphism among chains and inversions. Graph theory is

extensively used for structural Analysis and synthesis of Planar Kinematic Chains.

Some studies using graph theory are available on the isomorphism of Planetary

Gear Trains. Though the gears have lion share in transmitting motion it is felt that

it has not received as much attention as linkages and hence the desire to explore

this area more in detail. Another feature that prompted this study is that even to

date there is no convergence on the number of gear trains generated with 7 & 8

elements.

All the listed or available studies on PGTs pertain to synthesis of

PGTs with two major differences, The first one in the method used for generation

of PGTs and the second one in the test used for checking isomorphism in the

generated chain. However all the pertinent study available so far will not have

much significance if quantitative methods are not developed to compare all the

distinct gear trains with the same number of elements and DOF for different

aspects. Also it is always much desirable to anticipate the behaviour of the gear

trains without having to actually design, fabricate and test them. At present

designer depends on his intuition to select the best possible gear train and this may

not always lead to the optimum solution.

36

In this work besides giving a method for generation of epicyclic gear

trains using Hamming number method, a new method based on Moment concept

developed by Rao [53] for linkages is extended to PGTs and the same is also used

for testing isomorphism in PGTs generated along with Hamming number

Technique. Also moment concept is used to estimate relatively the aspect of

compactness in PGTs. Using Hamming number Method structural characteristics

of the PGTs like symmetry in PGTs, parallelism, pseudo isomorphism, rigidity,

greater speed ratios are studied.

Information theory applied earlier to kinematic chain is adapted to

PGTs and rating of EGTs is done based on their structures right at the design

stage. The aspects studied are transmission capacity power circulation and power

transmission efficiency.

*********

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